Crucial transistor performance information for amplifier design is often gathered through an extensive process that includes nonlinear measurement and characterization techniques, such as load-pull measurements. Performing load-pull measurements to characterize the maximum power, efficiency, or linearity performance of a device over changing load impedance often requires the measurement of many preselected impedance points. To reduce the number of measurements required in the design process, designers often rely on software nonlinear device models created from measurements over a subset of possible device configurations. Meanwhile, in reconfigurable power amplifiers, the optimum load impedance must be found through real-time measurements, which must be minimized to allow fast adaptivity. Both traditional and real-time design scenarios can benefit from techniques that are able to extract more information from fewer data points, either by reducing the size of load-pull measurement datasets, accelerating model development, or improving the rate of real-time optimization.

The ability of neural networks to learn and extrapolate performance of an individual large-signal microwave device using a measured dataset has been previously demonstrated [1], [2], [3], [4], [5], [6], [7]. Such approaches often seek to train an individual network to model a single device for simulation and design purposes, accelerating system development. However, given the typical computational cost associated with training neural networks, these methods may struggle in application to real-time optimization problems.

Instead of using neural networks, many other approaches for impedance optimization have been explored in the literature, including locating constant power contours and proceeding toward increased performance [8], fitting measured data to an equation and solving for the optimal value [9], genetic algorithms controlling a hybrid varactor/MEMS switch tuner [10], and alternatively searching for optimal reflection phase and gradually increasing reflection magnitude [11]. However, Barkate et al. [12] demonstrated that gradient-based methods perform well for real-time circuit optimization when compared with other typical algorithms.

Rather than train neural networks to model a device (i.e., predict the output of a device given some input), it is possible to instead train a network to directly predict measured performance characteristics, such as output power or gain, from sample measurements at a small number of configurations. As an example, consider the proposed problem shown in Fig. 1. Here, a subset of a device’s load impedance gain contours has already been evaluated, and it is desired to obtain the complete set of load-pull contours without additional measurement.

Fig. 1.

Illustration of load-pull extrapolation objective: given the partial set of load-pull measurements (top), it is desired to produce the complete set of contours (bottom) without additional measurement. Reprinted from [17].

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Saini et al. [13] previously demonstrated a form of load-pull extrapolation by generating polyharmonic distortion behavioral models [14] for the device in question, using as few as 15 measurements. It is possible to reduce the required information even further by instead applying image completion techniques from deep learning to extrapolate the unknown portions of the load-pull contours, as shown in Fig. 1. In general, image completion techniques have been successfully applied to complicated datasets, including full-color photographs [15] and partial electron microscopy [16].

In a preceding conference paper, we demonstrated the application of image completion for the simple case of simulated linear amplifier $S$ -parameter data [17], with good predicative capability shown using as few as nine measurements near the center of the Smith chart. This present article demonstrates, unlike the conference paper [17], success at extrapolating nonlinear load-pull data using the image completion method trained on similar generated linear $S$ -parameter data. The previous conference paper [17] used only generated $S$ -parameter data to simulate contours and attempt to predict the linear, simulated results with image completion. This present article demonstrates the accurate prediction of measured, nonlinear device contours. The successful application of this method to nonlinear devices demonstrated using multiple nonlinear load-pull measurements is, as such, a much broader and applicable accomplishment than the scope of [17]. In addition, this work provides a process in which a small number of points are algorithmically and iteratively selected during measurement to provide quality extrapolation with a minimum number of measurements, improving on the use of fixed, square sets of measurements from [17] and establishing how these techniques could be employed for live circuit optimization in an adaptive system. Finally, this present article demonstrates the successful use of the extrapolation approach in measurement results for a nonlinear amplifier, while work [17] is limited to simulation demonstration of extrapolation for generated linear amplifier $S$ -parameter models.

Section II provides an overview of the load-pull extrapolation method providing the basis for this work and introduces a novel family of circuit optimization algorithms built upon the iterative application of load-pull extrapolation. Section III then shows the performance of this optimization algorithm for live measurement data and compares the results to an existing, gradient-based method. Section IV provides a more detailed look at the quality of the individual extrapolations completed in the experiments of Section III. Finally, Section V presents conclusions from this work.

The load-pull extrapolation and circuit optimization techniques here are built upon the deep image completion method demonstrated for simulated, linear devices in [17]. For purposes of clarity, here is a brief overview of the approach.

To extrapolate full load-pull contours from a partial dataset, performance measurements from the known impedances are treated as an image for the purposes of image completion. The term “image completion” is derivative of the use of this approach to images. The “images” here correspond to performance data plotted on the Smith chart, while the “image completion” corresponds to filling in unknown portions of these data.

The image completion process applies a gradient search to a generative adversarial network (GAN) trained on known load-pull contours. GANs utilize two networks pitted against each other with opposing goals [18]. In this present article, the discriminator network is trained to determine whether an image is a member of the training set of load-pull contours or a false, generated image. Meanwhile, the generator network is trained to produce load-pull images that meet the criteria learned by the discriminator, resulting in misclassification. By iteratively training these networks (and taking care to avoid overfitting), the discriminator learns how to recognize valid load-pull contours and the generator learns how to produce valid load-pull contours. The operation of these networks is shown in Fig. 2, and the relationship of these networks to one another is described in Fig. 3.

Fig. 2.

Illustration of the function of the generator and discriminator networks utilized within the GAN architecture used in this present article.

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Fig. 3.

Illustration of the image completion search process used in this present article. The input to the generator network is revised over time to produce load-pull contours that have good agreement with the known performance contours and present the appearance of a real load-pull dataset.

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Given a well-trained GAN, load-pull extrapolation can be achieved by performing a gradient search on the inputs to the generator network, searching for an input $\hat {z}$ that produces a generated image $G(\hat {z})$ that closely agrees with the performance at the measured impedances. This is shown in Fig. 3. The method avoids the need to train the neural networks for the specific device under test, unlike the method used by Avolio et al. [1], which uses measurement data to train a network for modeling a specific device.

We use a Wasserstein GAN with gradient penalty (WGAN-GP). The Wasserstein GAN architecture reduces loss metric saturation, provides a more stable and robust training process, and is less susceptible to vanishing gradients and mode collapse [15], [19]. Because the WGAN-GP architecture uses the Wasserstein distance for quantifying how far the generated image is from the expected set (as opposed to categorizing the generated image as belonging to the set), the discriminator network is typically referred to as the “critic” network instead. The network implementation and layer topology for this present article are the same as in [17], originally adapted from [20]. Figs. 4 and 5 show the illustrations of the resulting network structure for the WGAN-GP generator and critic networks used in this present article.

Fig. 4.

WGAN-GP network topology for load-pull critic network. This network operates on a 32 $\times32$ pixel grayscale load-pull image and produces a single value estimating how far the input image is from the learned set of valid load-pull images. Illustration synthesized using [21].

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Fig. 5.

WGAN-GP network topology for load-pull generator network. This network operates on a 128-element initialization vector and produces a synthesized 32 $\times32$ pixel grayscale load-pull image. A hyperbolic tangent activation layer provides the final scaling of the network output. Illustration synthesized using [21].

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These networks have been trained using 100 000 sets of simulated output power contours for linear amplifier devices [17], represented as 32 $\times32$ pixel images. These data have been obtained by randomly generating amplifier $S$ -parameters and solving for the output power with various load impedances and a fixed 50-$\Omega$ source impedance using the transducer gain relationship from [22]. Each output power value is also subject to simulated measurement noise of up to $+720\,\,\mu \text{W}$ , which corresponds to about 0.1 dB of error at 15 dBm. The resulting images are standardized and preprocessed with a hyperbolic tangent function as recommended by Li et al. [23], matching the output form of the generator network.

For image completion, this present article uses a slightly modified version of the algorithm in [17], which is originally derived from [24]. This approach provides known data, specified as a masked image $\hat {x}$ , and produces a completed image containing valid load-pull contours that accurately reflect the known data. The mask of points providing measured inputs to the generator network is described [17] by

View Source\begin{align*} M\left ({n }\right)=\begin{cases} 1,& \hat {x}\left ({n }\right) \text {is valid} \\ 0,& \hat {x}\left ({n }\right) \text {is invalid} \\ \end{cases}\tag{1}\end{align*}

In comparing $G(\hat {z})$ the image that is generated by the generator network, to the measurement points, loss metrics are used. Contextual loss is the similarity between the generated image and the measured points $\hat {x}$ , using the mask $M$ , and is given by [17]

View Source\begin{equation*} L_{\mathrm {context}}\left ({\hat {z} }\right)=\left |{ \left |{ M\ast G\left ({\hat {z} }\right)-M\ast \hat {x} }\right | }\right |_{1}.\tag{2}\end{equation*}

$$\begin{equation*} L_{\mathrm {percept}}\left ({\hat {z} }\right)=\log \left ({1-C\left ({G\left ({\hat {z} }\right) }\right) }\right).\tag{3}\end{equation*}$$ View Source\begin{equation*} L_{\mathrm {percept}}\left ({\hat {z} }\right)=\log \left ({1-C\left ({G\left ({\hat {z} }\right) }\right) }\right).\tag{3}\end{equation*}

$$\begin{equation*} L\left ({\hat {z} }\right)=L_{\mathrm {context}}\left ({\hat {z} }\right)+\lambda L_{\mathrm {percept}}\left ({\hat {z} }\right).\tag{4}\end{equation*}$$ View Source\begin{equation*} L\left ({\hat {z} }\right)=L_{\mathrm {context}}\left ({\hat {z} }\right)+\lambda L_{\mathrm {percept}}\left ({\hat {z} }\right).\tag{4}\end{equation*}

For loss minimization, we use the Adam optimization [25] with learning rate 0.01, $\beta _{1}= 0.9$ , $\beta _{2}= 0.999$ , and $\epsilon ={10}^{-8}$ . While work [17] uses a fixed number of optimization iterations when searching for an appropriate generator network input, this present article instead monitors the loss metrics used during optimization, greatly reducing the computation required for each extrapolation. When the combined loss metric reaches a local minimum, the search concludes, and the corresponding generator output provides the load-pull extrapolation result.

The general process flow for a circuit optimization algorithm using only load-pull extrapolation to drive exploration of the search space is shown in Fig. 6. For each of the given steps in the algorithm, different variations are possible.

Fig. 6.

Generic process for circuit optimization using load-pull extrapolation. Variations on the algorithm are possible by changing how the initial and subsequent measurement points are selected based on extrapolation results and by changing the convergence criteria.

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The algorithm implementation used in this present article is built on a simple “maximum addition” technique, where the best predicted impedance from the previous load-pull extrapolation is chosen for measurement, and its performance is added to the dataset used for the next extrapolation. The initial point is chosen at random from one of the four pixels closest to 50 $\Omega$ . Using this measurement, an (essentially random) optimum is generated by extrapolation, and this optimum is then measured and used with the measurement at the initial 50-$\Omega$ point to predict the next optimum. The predicted optima continue to be added to the dataset of measured points, and the search converges when the most recently predicted best impedance has already been evaluated during a previous iteration, at which point the best measured impedance is reported as the optimum. If applied to a frequency hopping context or live system, the current impedance could be used as the initial point of the optimization process instead of first tuning near 50 $\Omega$ .

This maximum addition technique is well suited for systems where the time cost of performing a load-pull extrapolation is low relative to the time required to tune to and evaluate a new impedance value. However, if this situation is reversed (i.e., it is faster to tune and measure the performance extrapolations), it may be beneficial to evaluate multiple impedances per extrapolation. For example, the eight pixels surrounding a predicted optimal impedance could also be evaluated in addition to the optimal impedance. This also permits a more sophisticated convergence check that looks for the presence of some local maximum performance. Given the typical convex nature of output power contours on the Smith chart, any local maximum would be assumed to be the global maximum without requiring an additional search.

Following up on the results mentioned in [17] for linear devices, this section considers the quality of the load-pull extrapolations performed as part of the searches in Section III. Over the course of those searches, extrapolations were performed using anywhere from 1 to 14 measurements. However, only one extrapolation each used 13 and 14 measurements, and only five extrapolations used 12 measurements. Figs. 13 and 14 show statistics for the error in the predicted power and optimal impedance for all extrapolations with at least ten extrapolations per input dataset size. Table I provides a numeric summary of these results for comparison with the simulated, linear results mentioned in [17].

TABLE I Nonlinear Load-Pull Extrapolation Performance: Predicted Peak Statistics

Fig. 13.

Power error box plot (median, quartiles, and extremes) for the extrapolations performed in Section III. Here, an error is the difference between the measured load-pull maximum performance and the maximum power reported by the extrapolated load-pull.

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Fig. 14.

Load impedance error box plot (median, quartiles, and extremes) for the extrapolations performed in Section III. Here, an error is the distance between the measured load-pull optimal reflection coefficient and the optimal reflection coefficient reported by the extrapolated load-pull.

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Comparing the results mentioned in Table I and [17], the extrapolations based on at least seven measurements perform comparably to the linear results with 81 measurements grouped around the center of the Smith chart, based on the median $\Gamma _{L}$ error magnitude. (For comparison purposes, we prefer the median statistic as it more closely represents the typical extrapolation result without as much influence from the rare strong outliers visible in Figs. 13 and 14.) This finding suggests that the load-pull extrapolation benefits greatly from having a collection of points scattered throughout the Smith chart, as opposed to a group of points near a single area. This finding may be useful for other microwave device modeling applications, such as the results mentioned in [1], which used measurements from a uniform grid in a single area of the Smith chart. In addition, the extrapolation method continues to perform quite well on nonlinear measurements, despite having only been trained on simulated, linear devices.

The ability to extrapolate nonlinear characteristics of amplifier output power versus reflection coefficient (load-pull) from partial datasets utilizing deep learning image completion techniques has been demonstrated. This approach has produced good predictive ability (median optimal $\Gamma _{L}$ error vector magnitude below 0.1) with as few as seven measured impedances, even when the underlying neural networks are only trained on randomly generated linear $S$ -parameters. Load-pull extrapolation accelerates amplifier optimization processes, such as traditional device characterization load-pull measurements or real-time impedance tuning techniques, by quickly identifying areas of interest. Accelerated impedance tuning has been demonstrated using an iterative extrapolation method, providing significant time savings over a gradient search method using the same hardware measurement platform.